Answer:
The value is [tex]P( X < 0.09) = 0.070781[/tex]
Step-by-step explanation:
From the question we are told that
The population proportion is p = 0.11
The sample size is n = 529
Generally given that the sample size is large enough (i.e n > 30), then the mean of this sampling distribution is mathematically represented as
[tex]\mu_{x} = p = 0.11[/tex]
Generally the standard deviation of this sampling distribution is
[tex]\sigma = \sqrt{ \frac{ p (1 - p )}{n} }[/tex]
=> [tex]\sigma = \sqrt{ \frac{ 0.11 (1 - 0.11 )}{529} }[/tex]
=> [tex]\sigma = 0.01360[/tex]
Generally the probability that the proportion of defective bottles in a sample of 529 bottles would be less than 9% (0.09) is mathematically represented as
[tex]P( X < 0.09) = P(\frac{X - \mu_{x}}{\sigma} < \frac{0.09 -0.11}{ 0.01360} )[/tex]
[tex]\frac{X -\mu}{\sigma } = Z (The \ standardized \ value\ of \ X )[/tex]
[tex]P( X < 0.09) = P(Z< -1.47 )[/tex]
Generally from the z table the area under the normal curve to the left corresponding to -1.47 is
[tex]P(Z< -1.47 ) = 0.070781[/tex]
So
[tex]P( X < 0.09) = 0.070781[/tex]
